Triangulations of cyclic polytopes from the point of view of cluster theory

I will describe the set of triangulations of a 2d-dimensional cyclic polytope in a way that naturally generalizes the description of triangulations of a n-gon in terms of a collection of non-crossing diagonals. The diagonals of a polygon correspond to the cluster variables in the simplest family of cluster algebras, and there are certain easily-described relations among the associated variables. Tropicalizing these relations gives rise to relations which have an interpretation in terms of counting intersections with laminations, as in the work of Gekhtman-Shapiro-Vainshtein and Fomin-Shapiro-Thurston. We give similar tropical relations in the 2d-dimensional setting, with a similar interpretation in terms of counting intersections with a lamination. The problem of finding a meaningful detropicalization of these relations is still open. This is joint work with Steffen Oppermann, based on arXiv:1001.5437